# Definite integrals and area

1. Feb 28, 2015

### titasB

1. The problem statement, all variables and given/known data

Find ∫ f(x) dx between [4,8]

if,

∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4

2. Relevant Equations

∫ f(x) dx between [4,8] ,
∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4

3. The attempt at a solution

We are given ∫ f(2x) dx between [1,4] = 3

Let, 2x = u ⇒ dx = du/2

So, the new intervals are u2 = 2(4) = 8 and u1 = 2(1) = 2

This gives: 1/2 ∫ f(u) du between [2,8] = 3 ⇒ ∫ f(u) du between [2,8] = 6

And so to find t ∫ f(x) dx between [4,8]

I subtract ∫ f(x) dx between [2,4] from ∫ f(2x) dx between [1,4]

which is the same as writing: ∫ f(u) du between [2,8] - ∫ f(x) dx between [2,4] = 6- 4 = 2

Is this the correct answer? I'm not sure if ∫ f(u) du between [2,8] = 6 is the same as ∫ f(x) dx between [2,8] = 6
I read something about a dummy variable and this seems like a reasonable answer. Please let me know.

2. Feb 28, 2015

### wabbit

Yes your reasoning and result are correct. And indeed x or u are just placeholders in the integrals, you can replace them with any symbol you like.

3. Feb 28, 2015

thank you